Tube domain and an orbit of a complex triangular group

Hideyuki Ishi, Takaaki Nomura

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8 Citations (Scopus)


Let w be a complex symmetric matrix of order r, and Δ 1(w), . . . , Δ r (w) the principal minors of w. If w belongs to the Siegel right half-space, then it is known that Re (Δ k (w)/Δ k-1(w)) > 0 for k = 1, . . . , r. In this paper we study this property in three directions. First we show that this holds for general symmetric right half-spaces. Second we present a series of non-symmetric right half-spaces with this property. We note that case-by-case verifications up to dimension 10 tell us that there is only one such irreducible non-symmetric tube domain. The proof of the property reduces to two lemmas. One is entirely generalized to non-symmetric cases as we prove in this paper. This is the third direction. As a byproduct of our study, we show that the basic relative invariants associated to a homogeneous regular open convex cone Ω studied earlier by the first author are characterized as the irreducible factors of the determinant of right multiplication operators in the complexification of the clan associated to Ω.

Original languageEnglish
Pages (from-to)697-711
Number of pages15
JournalMathematische Zeitschrift
Issue number3
Publication statusPublished - Jul 2008

All Science Journal Classification (ASJC) codes

  • Mathematics(all)


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